Question: $f(x, y) = x^3 + 5x^2y$ What is $\text{div}(\text{grad}(f))$ ? $\text{div}(\text{grad}(f)) = $
Answer: The Laplacian of a scalar field $f$ is the sum of each of its second partial derivatives. $\text{div}(\text{grad}(f)) = \dfrac{\partial^2 f}{\partial x^2} + \dfrac{\partial^2 f}{\partial y^2} + \dfrac{\partial^2 f}{\partial z^2}$ [What does it mean to take the divergence of the gradient?] Let's find the second partial derivatives of $f$ ! $\begin{aligned} f_{xx} &= \dfrac{\partial}{\partial x} \left[ \dfrac{\partial f}{\partial x} \right] \\ \\ &= \dfrac{\partial}{\partial x} \left[ 3x^2 + 10xy \right] \\ \\ &= 6x + 10y \\ \\ f_{yy} &= \dfrac{\partial}{\partial y} \left[ \dfrac{\partial f}{\partial y} \right] \\ \\ &= \dfrac{\partial}{\partial y} \left[ 5x^2 \right] \\ \\ &= 0 \end{aligned}$ The Laplacian is $\text{div}(\text{grad}(f)) = f_{xx} + f_{yy}$. Therefore: $\text{div}(\text{grad}(f)) = 6x + 10y$